A Location Allocation Model for a Territorial Design Problem with Dense Demand

A Location Allocation Model for a Territorial Design Problem with Dense Demand

María Beatríz Bernábe Loranca (Faculty of Computer Science, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico), Rogelio González Velázquez (Faculty of Computer Science, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico), Elias Olivares Benítez (Interdisciplinary Center for Graduate Studies and Research, Universidad Popular Autónoma del Estado de Puebla, Puebla, Mexico) and José Luis Martínez Flores (Interdisciplinary Center for Graduate Studies and Research, Universidad Popular Autónoma del Estado de Puebla, Puebla, Mexico)
Copyright: © 2016 |Pages: 14
DOI: 10.4018/IJAL.2016010101
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The authors present in this work a proposal that allows establishing the relationships between the facilities location problem and the client allocation within a dense demand environment in territorial design. This proposal can be seen as a basic methodology to give support to the decision making. The use of this application lets one know the production facilities location, warehouses or distribution centers in a geographical space division. On the other hand, solving the client's dense demand for goods or services, means finding the location of facilities in a populated geographic territory, where the population has a demand for services in a constant basis. Finding the location is obtaining the decimal geographical coordinates in longitude, latitude where the facility is located, such that the product or service transport costs the least. The implications and practical benefits of the results from this work allow an organization to be able to design an efficient logistics plan in benefit of its supply chain. The problem the authors present is a territorial design and optimization one due to the fact that the required territorial partition demands the creation of compact zones where the minimum distance between the geographical objects is implicitly optimized. This problem belongs to the NP-hard class, where the use of a metaheuristic to attain approximated solutions becomes a necessity, Variable Neighborhood Search was chosen because it achieves good solutions for this kind of problems. Once the territory has been partitioned into k zones where the centroid of each zone is the distribution center, as a first step the authors proceed to apply the dense demand with continuous functions and this is the main original contribution of this paper: Finding the location of facilities inside a territory where the population has a dense demand for services. The location obtained means having available the decimal geographic coordinates in longitude and latitude from the location point in such a way that the products or services transfer has a minimum cost. Finally the Weber function is minimized, which weights are a function that represents the population's demand in every territory, multiplied by the Euclidean distance between the potential location points and demand points.
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1. Introduction

We understand Territorial Design (TD) or zone design as the grouping of small geographical areas or units into zones that are acceptable according to the requirements imposed by the problem under study. Depending on the context, these requirements can include the generation of connected zones that contain about the same amount of habitants, clients, communication means, public services, etc. The zone design has several applications such as the creation of school zones, zones with the appropriate characteristics for socioeconomic analysis, sales, maintenance or services territories design, geographical design for census and logistic applications. Diverse interesting TD applications can be seen in Ricca et al. (2008), Rios-Mercado et al. (2007), Salazar et al. (2011) and Salazar et al. (2011a). In the geographical aggregation process in TD, hierarchical or partitioning clustering methods adequate to the problem are often employed.

For the aggregation done in this work, the elements grouped belong to the geographical territorial unit known as Basic Geo-statistical Area (AGEB for its initials in Spanish), which is defined as the minimum geographical division used for census and statistical purposes in the National Institute of Geography and Informatics INEGI for its initials in Spanish (INEGI, 2011). INEGI is an autonomous entity of the Mexican government, dedicated to the coordination of the National Informatics Statistics and Geography System and is the institution in charge to do the population census every ten years. Its main goal is to achieve the provision of opportune, verified, relevant and quality information to the public and the State, with the aim of contributing to the national development, under the accessibility, transparency, objectivity and independency principles (INEGI, 2011).

The groups generated by the partitioning we have developed are formed by a set of AGEBs and due to the partitioning properties; the compactness characteristic of TD is implicitly satisfied. This territorial partitioning is previously necessary to solve the location allocation problem (LAP) for a TDP with dense demand, which is the main purpose of this work: finding the geographical coordinates (longitude, latitude) of the location of a distribution center (DC) that provides services to a group of communities that are found in every AGEB, where each AGEB is represented by its centroid. The location should be the one that minimizes the travelling expenses by finding the geographical coordinates of the center of the centroids. The populations from these communities represent the potential clients of the DC, and their demand is modeled continuous demand functions with two variables based on population density of every group (Newling, 1969).

Due to the numeric nature of the solutions obtained, this problem addresses a continuous case of LAP. Additionally, with the mathematical approach associated, we make use of a geographical information system (GIS) with the goal of creating maps (Zamora, 2006).

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